In June, Luis Alberto Cuan Lio and his pregnant wife, Yordana Bravo Perez, flew from Cuba to Ecuador as tourists. It was the first leg in a circuitous journey that ended when they crossed the border from Mexico into the U.S., where they are building a new life.

An influx of illegal immigrants from Central America drew wide attention recently, but more than 22,000 Cubans entered the U.S. over land in the year that ended Sept. 30, twice as many as in the previous year. An additional 3,940 sought to reach the U.S. along maritime routes, nearly double the previous year and the highest number since 2008, when the island was buffeted by several hurricanes, exports were suffering and Raúl Castro became president.

Helping to fuel the exodus are the Cuban government’s easing of travel restrictions for its citizens last year and a general lack of hope among Cubans for the country’s economic prospects. “People are growing frustrated with the depth and pace of the economic reforms,” said Ted Henken, a Latin American studies professor at Baruch College in New York.

The U.S. also has loosened restrictions on visas for Cubans who arrive by air as tourists, some of whom are believed to remain in the U.S. About 30,000 of these visas were issued in the fiscal year ended Sept. 30, 2013.

Cuban migrants to the U.S. enjoy special treatment. The Cold War-era Cuban Adjustment Act of 1966 allows those who touch U.S. soil to remain here rather than be deported. They are eligible for some benefits that are accorded to refugees fleeing persecution. After a year, they can apply for permanent residency or a green card.

The number of new Cuban arrivals pales next to the 130,000 or so Central Americans that crossed into the U.S. illegally in the latest fiscal year, or the estimated 125,000 Cubans that came by sea in 1980’s Mariel boatlift. But the land crossings open a new chapter in Cuban flight to the U.S., and experts expect arrivals to keep climbing.

At the Mexico-Texas border, Cubans join the line for people with permission to enter the U.S. “They know exactly where to go, arrive with their documents and say they want to apply for the Cuban Adjustment Act,” said Adriana Arce, assistant director at the Laredo port of entry for U.S. Customs and Border Protection, who has had to bolster staffing to process the influx. After an interview, the Cubans typically are “paroled” into the country, a process that takes roughly two hours. Then, armed with U.S.-issued papers, they are free to go wherever they wish.

Mr. Cuan, a physician, and his wife, a preschool teacher, saved money for years and used help from friends and relatives in the U.S. to pay for their journey to the U.S., which cost about $8,000.

“We had lost hope of Cuba improving,” said the 47-year-old Mr. Cuan, who says he earned the equivalent of $20 a month in Cuba. Rather than risk their lives at sea, they opted for a safer, if longer and pricier, route.

The couple flew to Ecuador because it didn’t require a visa. From there, they traveled to Lima, Peru, where they secured fake Peruvian passports that enabled them to enter Mexico without a visa. In Monterrey, they boarded a bus to Laredo, where they told U.S. inspectors that they were Cuban nationals.

The couple is now in Houston, where they are being helped by Refugee Services of Texas. That nonprofit agency over the summer handled as many Cubans as it did refugees from all other countries combined. “It’s a lot of people all at once,” said Sara Kauffman, area director.

Wafa Abdin, a vice president of Catholic Charities of the Archdiocese of Galveston-Houston, says the agency has been serving about 50 Cubans a week this year, up from about five to 15 a week two years ago. “Our staff is used to helping everyone who comes through the door; this is nonfunded need,” that is stretching resources, said Margaret Ayot, a program supervisor.

YMCA International Services of Houston served 700 refugees from Africa, Asia and other countries in the 2014 fiscal year. In that same period, it provided assistance to 434 Cubans. “They present themselves and we piece together a plan quickly,” said executive director Jeff Watkins.

Cubans are eligible for eight months of cash assistance, medical coverage, job-placement services and free English classes, among other benefits offered to refugees. For a family of three, eight months of cash assistance totals $4,300; for a single person, it is about $2,500. Cubans can also receive food stamps if they meet the income requirements.

Some say Cubans aren’t refugees in the traditional sense, though they have such rights. “The majority of Cubans are economic migrants,” said Jodi Goodwin, an immigration attorney who works along the Texas border. The U.S. admitted 70,000 refugees last year, mainly from Africa, Asia and the Middle East.

Cubans who take to the sea to reach Miami are typically the poorest, traveling on vessels made with rebar, wood and Styrofoam, according to Capt. Mark Fedor, chief of response for the Coast Guard’s Seventh District in Miami. The spate of recent crossings is causing alarm, he said. “It’s sometimes ingenious what they put together,” he said. “But they’re very unseaworthy.”

The Cuban influx is expected to accelerate, experts say, especially if the U.S. economy keeps improving.

Mr. Cuan says he is looking for work in any field after receiving his employment authorization and a Social Security number. “We have nothing here,” he said, but “it’s a better place to live.”